Taylor series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \sin x=?\,}

This article/section deals with mathematical concepts appropriate for late high school or early college.

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , the Taylor series about the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=f(x_0)+(x-x_0)\frac{df(x)}{dx}+\frac{(x-x_0)^2}{2!}\frac{d^2f(x)}{dx^2}+\ldots+\frac{(x-x_0)^N}{N!}\frac{d^Nf(x)}{dx^N}}

where each of the derivatives is to be evaluated at $x=x_{0}$. If as $N\rightarrow \infty$ the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around $x_{0}=0$, in which case they are sometimes also known as a Maclaurin series.

Examples of common Taylor series

$e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\ldots$

$\sin x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{(2n+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\ldots$

$\cos x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n}}{(2n)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\ldots$

Extensions of the Exponential Function

Consider the exponential of imaginary number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yi} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{yi}=1+yi+\frac{(yi)^2}{2!}+\frac{(yi)^3}{3!}+\frac{(yi)^4}{4!}+\frac{(yi)^5}{5!}+\ldots}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =1+yi-\frac{y^2}{2!}-i\frac{y^{3}}{3!}+\frac{y^4}{4!}+i\frac{y^5}{5!}-\ldots}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(1-\frac{y^2}{2!}+\frac{y^4}{4!}-\ldots)+i(y-\frac{y^{3}}{3!}+\frac{y^5}{5!}-\ldots)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\cos y+i\sin y}

By the power laws then all complex numbers have an exponential,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{x+yi}=e^{x}(\cos y+i\sin y)} .